Solving Equations with One Variable using One Function

As we have seen, we can solve an equation $f(x)=0$ by finding the $x$-intercepts of its function $y=f(x)$, and the graph of $y=f(x)$ makes the process much easier. For example, we can easily locate the $x$-intercept of $f(x)=-\frac{3}{4}x+3$ from its graph, and we can conclude that the solution of $-\frac{3}{4}x+3=0$ is $x=4$.

Figure 3. Graph of $f(x)=-\frac{3}{4}x+3$

Moreover, this method helps us to solve more complex equations as well. For example, to solve $x+\frac{3}{x}-4=0$, we should consider the graph of $f(x)=x+\frac{3}{x}-4$ and find the $x$-intercepts. As we can read on the graph, the $x$-intercepts of this function are $(1, 0)$ and $(3, 0)$. So, the solutions of the equation are $x=1, 3$.

Figure 4. Graph of $f(x)=x+\frac{3}{x}-4$

Example: Solving $f(x)=0$ Graphically

Solve the equation graphically. Use a graphing tool.

$x^3-x-4x^2+4=0$

 

Show Graph

Show Answer

Let $f(x)=x^3-x-4x^2+4$. Then $x=-1, 1, 4$. 

Try It

Solve the equation graphically. Use a graphing tool.

$(x-1)^4-3(x-1)^2-4=0$

 

Show Graph

Show Answer

Let $f(x)=(x-1)^4-3(x-1)^2-4$. Then, $x=-1, 3$.

Example: Solving $f(x)=0$ Graphically

Solve the equation graphically. Use a graphing tool.

$\sqrt[3]{(x+1)^2}-4=0$

 

Show Graph

Show Answer

Let $f(x)=\sqrt[3]{(x+1)^2}-4$. Then, $x=-9, 7$

Try It

Solve the equation graphically. Use a graphing tool.

$\frac{x-3}{x}-\frac{1}{2}=0$

 

Show Graph

Show Answer

Let $f(x)=\frac{x-3}{x}-\frac{1}{2}$. Then, $x=6$

Solving Equations with One Variable using Multiple Functions

Now we know how to solve equations graphically. But what if the given equation is not in “$f(x)=0$” form? How can we apply the method we used in the previous examples? There are two different ways to solve this equation:

1. Move all terms to the left and make it in “$f(x)=0$” form. Then, graph and find the $x$-intercepts.
2. Let “$f(x)=$(lefthand side)” and “$g(x)=$(righthand side).” Then, graph and find the intersecting points of $f(x)$ and $g(x)$. The $x$-coordinates are the solutions of the equation.

Try It

Solve the equation graphically. Use a graphing tool.

$-2x^3+6x-3=x^2-3$

Show Answer

Move all terms to the lefthand side: $-2x^3+6x-3-x^2+3=0$. Then let $f(x)=-2x^3-x^2+6x$. Since the $x$-intercepts are $(-2, 0), (\frac{3}{2}, 0)$, the solutions are $x=-2, \frac{3}{2}$.

OR

Let $g(x)=-2x^3+6x-3$ and $h(x)=x^2-3$. Since the intersecting points of $g(x)$ and $h(x)$ are $(-2, 1)$ and $(\frac{3}{2}, -\frac{3}{4})$, the solutions are $x=-2, \frac{3}{2}$.